emre wrote:> Are you saying that a one-to-one > transformation, that is the Fourier transform (FT), of a random signal is > not random? But this is a contradiction, since there is only one function > (scaled delta) that is the (inverse) FT of a constant function.There's a bit of apples and oranges comparison here. The real part and the imaginary part of the FT of a random signal may very well be zero-mean random (but not necessarily normally distributed). But the OP asked about magnitude and phase. The magnitude cannot be zero mean, since it is everywhere nonnegative, and its distribution will depend upon the distribution exhibited by the real and imaginary parts. The phase can be zero mean, but it is unlikely that it will exhibit any distribution other than uniform. Greg
White noise generation in the frequency domain
Started by ●September 22, 2008
Reply by ●September 23, 20082008-09-23
Reply by ●September 23, 20082008-09-23
On Sep 23, 7:34�am, Greg Berchin <gberc...@sentientscience.com> wrote:> emre wrote: > > Are you saying that a one-to-one > > transformation, that is the Fourier transform (FT), of a random signal is > > not random? �But this is a contradiction, since there is only one function > > (scaled delta) that is the (inverse) FT of a constant function. > > There's a bit of apples and oranges comparison here. �The real part > and the imaginary part of the FT of a random signal may very well be > zero-mean random (but not necessarily normally distributed). �But the > OP asked about magnitude and phase. �The magnitude cannot be zero > mean, since it is everywhere nonnegative, and its distribution will > depend upon the distribution exhibited by the real and imaginary > parts. �The phase can be zero mean, but it is unlikely that it will > exhibit any distribution other than uniform.If both the real and imaginary parts are Gaussian, the magnitude will be Rayleigh. Jerry
Reply by ●September 23, 20082008-09-23
>If it is not constant magnitude, then I think you have a dilemma: >at what frequencies would it have above-average magnitude? >Logically there can't be any.As Jerry just pointed, if the real and imaginary parts are independent Gaussian (with same variance), then the magnitude is Rayleigh distributed random. It is not constant. The precise statement should be that the average (expected value) of the magnitude is constant over frequencies. This is the characteristic of white noise. I believe this is what you really meant. Emre
Reply by ●September 23, 20082008-09-23
>Sorry but I don't use Matlab (I code all in C) and my mathematics >background is a bit weak. What do you mean by "sigma" in the context >of "sigma/sqrt(2)"? Does it have anything to do with the summation >operator?sigma (the Greek letter written in small -- not capital as in summation operator) is usually used to denote the standard variation (the square root of the variance) of a normal (Gaussian) random variable.>Also, if you generate the real and imaginary parts separately, >wouldn't the distribution of the samples in the complex plane be a >square?Close. It is called "circularly symmetric". It is interesting, but it is the case that real and imaginary parts of the noise turn out to be independent in many cases, most prominently for thermal noise. If you are trying to simulate thermal noise in your measurement device, then Gaussian distribution is likely the right thing to use. If your original signal is real, then you have to take care about the generation directly in the frequency domain, because the FT of a real signal has conjugate symmetry: you should generate half of the samples (say, those corresponding to positive frequency indices,) and set the other half to the according the the conjugate symmetry. You can refer to any good signal processing book for this property. If you would like to catch up on these things, I believe you could use Schaum's Outlines, on Probability, Random Variables, and Random Processes, and Digital Signal Processing. They are both affordable and easy to follow. Hope this helps. Emre
Reply by ●September 23, 20082008-09-23
>If you would like to catch up on these things, I believe you could use >Schaum's Outlines, on Probability, Random Variables, and RandomProcesses,>and Digital Signal Processing. They are both affordable and easy to >follow.Actually, I haven't read the latter one above, but people seem to be happy with Schaum's Outline of Signal Processing (not digital, there is no review on that at Amazon.) Depending on the time you can commit, I would also recommend other signal processing texts, such as Rich Lyons' for starters, and possibly Oppenheim's after that... As far as the topic of Probability is concerned, Schaum's Outline may indeed be a good starting point. You might later choose to read H. Stark's "Probability, Random Processes, and Estimation Theory". Emre
Reply by ●September 23, 20082008-09-23
>recommend other signal processing texts, such as Rich Lyons' forstarters, I meant Richard Lyons.. Sorry.
Reply by ●September 23, 20082008-09-23
On Mon, 22 Sep 2008 07:31:56 -0700, Greg Berchin wrote:> Vladimir Vassilevsky wrote: > >> This is not right. That way are generating a sum of the synchronously >> phase manipulated signals with the breaks of phase at the same point. >> there will be the artifacts at the edges of the subsequent FFT blocks, >> and you will see it in the frequency domain, too. You have to make >> random magnitude in the addition to the random phase. > > I think that your description of the problem is right but your solution > is wrong. The resulting "white" noise will be pseudorandom with period > equal to the FFT size. But using a random FFT magnitude won't change > that. The Fourier Transform of white noise has constant magnitude and > random phase. > > GregNuh uh. The _expected_ value of the Fourier transform of white noise has constant magnitude and random phase, but not the _actual_ value. To choose a simple example, the FFT of a series of zero-mean, iid Gaussian samples will have bins whose complex and real parts are all iid Gaussian variables. But they will _not_ have constant magnitude. This concept extends to the continuous time Fourier transform, but then if you choose to start with really white noise you have to wrestle with all the infinities. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Reply by ●September 23, 20082008-09-23
On Mon, 22 Sep 2008 19:16:44 -0400, Greg Berchin wrote:> On Mon, 22 Sep 2008 20:44:46 +0000 (UTC), spope33@speedymail.org (Steve > Pope) wrote: > >>Whereas the values of the discrete Fourier transform of a white noise >>signal have a normal distribution. > > I'm not doubting the statements from you and Vladimir, but I've never > seen the derivation of this. Where did you find this info? > > GregWell, each bin of the FFT is a weighted sum of the original sample -- only the weights have a special relationship to the weights of the other bins. So, the weighted sum of a series of iid Gaussian variables is... -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Reply by ●September 23, 20082008-09-23
On Mon, 22 Sep 2008 16:51:54 -0700, Jerry Avins wrote:> On Sep 22, 7:25 pm, spop...@speedymail.org (Steve Pope) wrote: >> Greg Berchin <gberc...@comicast.net> wrote: >> >> >On Mon, 22 Sep 2008 20:44:46 +0000 (UTC), spop...@speedymail.org >> >>Whereas the values of the discrete Fourier transform of a white noise >> >>signal have a normal distribution. >> >I'm not doubting the statements from you and Vladimir, but I've never >> >seen the derivation of this. Where did you find this info? >> >> Let's see... the DFT is a linear operation, that is to say, any given >> output of a DFT is a linear combination of the inputs to the DFT, so if >> the inputs are all normal, then any output is normal, since the sum of >> normal variables is normal. >> >> This applies to both the real part of an output, and the imaginary part >> of an output. They would each be normal. > > Noise needn't be bormally distributed in order to be white. Why would > its parts be Gaussian? > > JerryGood point. But I'll bet that most samples, after "fft-izing" will look Gaussian. If they don't you could use a longer sample (for stationary noise). Only pathological cases (like most accurate models of atmospheric noise at LF and MF) would have the requisite infinite variance to make this expedient less than speedy. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
